Block avoiding point sequencings of partial Steiner systems

Abstract

A partial(n,k,t)_lambda -system is a pair (X,{mathcal {B}}) where X is an n-set of vertices and {mathcal {B}} is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most lambda blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of {0,ldots ,n-1}. A sequencing is ell -block avoiding or, more briefly, ell -good if no block is contained in a set of ell vertices with consecutive labels. Here we give a short proof that, for fixed k, t and lambda , any partial (n,k,t)_lambda -system has an ell -good sequencing for some ell =Theta (n^{1/t}) as n becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case k=t+1 where results of Kostochka, Mubayi and Verstraëte show that the value of ell cannot be increased beyond Theta ((n log n)^{1/t}). A special case of our result shows that every partial Steiner triple system (partial (n,3,2)_1-system) has an ell -good sequencing for each positive integer ell leqslant 0.0908,n^{1/2}.